The 5.56, AKA .223 Remington, is 44.7 mm long in the case. Bullet weights range from 45 grains up to 80 grains. The ballistics depend on the type of weapon the round is fired from, the barrel length and twist rate, the bullet type and weight, the powder type and charge weight. There are other smaller variables, but these are the major elements which determine the ballistic behavior of the cartridge.
The 7.62, AKA .308 Winchester, is 51.2 mm long in the case. Bullet weights range from 110 grains up to 180 grains. There are heavier .308 bullets, but practically speaking, 180 grs is about the upper limit for the .308 in commercial and military loadings. The same factors apply to ballistic behavior of this round. A reloading manual will explain all.
A 5.56 round may or may not be higher in velocity than a 7.62 round. It depends on the factors stated above. The 5.56 does not, nor does any other cartridge, have a fixed set of ballistic characteristics. There are too many variables at loading and in shooting. Each of the mentioned cartridges can be loaded with the right bullet and fired with sufficient velocity to impart adequate energy depending on the bullet weight to penetrate level III armor.
Get it? It's all variable. No pat answers. Every answer is a set of conditions. So, you can't say this cartridge will do that and that one will do this, until you get to the extremes of their capabilities.
The big differences are: the .223 shoots a little flatter, the .308 shoots a heavier bullet, so takes energy farther down range. They both kill, but the .308 can do it farther from the muzzle.
Both are accurate and that counts for a whole lot more than which one can penetrate level III body armor. One shot, One kill.
"Get your mind right and the body will follow." - Shino Takazawa, sinsei, hachi dan, Keishinkan do.